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6.045J/18.400J: Automata, Computability and Complexity Nancy Lynch Homework 12 Due: May 7, 2007 Elena Grigorescu Readings: Section 9.1, 9.2 Problem 1 : Sipser 9.13 Let pad ( s, l ) be the function that adds enough copies of the symbol # to the end of the string s so that the length of the result is at least l . For any language A and function f : N N define the language pad ( A, f ( m )) = { pad ( s, f ( m )) | where s A, m is the length of s } . Prove that if A T IME ( n 6 ) then pad ( A, n 2 ) T IME ( n 3 ). Solution 1 : Consider the following decider for pad ( A, n 2 ): M:” on input x # t use the decider for A to decide if x A if no reject; if yes, check that t = | x | 2 − | x | and output yes if so and no otherwise.” M runs in time O ( N 3 ) since now the size of the input is N = n 2 . Problem 2 : Sipser 9.14 Prove that, if NEXPT IME negationslash = EXPT IME then P negationslash = NP . Solution 2 : Suppose NEXPT IME negationslash = EXPT IME . Then there exists a language B s.t. B is decidable in 2 p ( n ) time by a NTM (for some polynomial p), and B is not decidable in 2 q ( n
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